A numerical study of a particular non-conservative hyperbolic problem

We study the ability of several numerical schemes to solve a non-conservative hyperbolic system arising from a flow simulation of solid–liquid–gas slurries with the so-called virtual mass effect. Two classes of numerical schemes are used: some Roe-type finite volume schemes, which are based on the resolution of linearized Riemann problems, and some (centered or upwind) schemes with an additional artificial diffusion, such as the classical Rusanov scheme. For flow regimes of interest (steady as well as unsteady flows), the computational process breaks down for some schemes. Indeed, for such flows, the system has at least one eigenvalue having a small magnitude in the interior of the computational domain and this is a possible reason for the failure of some upwind schemes using the resolution of a linearized Riemann problem. Such a failure does not appear with, for instance, the Rusanov scheme which is well known for its robustness. Furthermore, since the system is non-conservative, it is not clear what a weak solution is, when the solution is discontinuous (at least, one needs to have the non-conservative equivalent of the Rankine–Hugoniot jump conditions) and we show that the approximate solution given by different numerical schemes converges towards different “weak solutions”.